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Theorem onsucuni2 6558
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onsucuni2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )

Proof of Theorem onsucuni2
StepHypRef Expression
1 eleq1 2526 . . . . . 6  |-  ( A  =  suc  B  -> 
( A  e.  On  <->  suc 
B  e.  On ) )
21biimpac 486 . . . . 5  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  B  e.  On )
3 eloni 4840 . . . . 5  |-  ( suc 
B  e.  On  ->  Ord 
suc  B )
4 ordsuc 6538 . . . . . . . 8  |-  ( Ord 
B  <->  Ord  suc  B )
5 ordunisuc 6556 . . . . . . . 8  |-  ( Ord 
B  ->  U. suc  B  =  B )
64, 5sylbir 213 . . . . . . 7  |-  ( Ord 
suc  B  ->  U. suc  B  =  B )
7 suceq 4895 . . . . . . 7  |-  ( U. suc  B  =  B  ->  suc  U. suc  B  =  suc  B )
86, 7syl 16 . . . . . 6  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  suc  B
)
9 ordunisuc 6556 . . . . . 6  |-  ( Ord 
suc  B  ->  U. suc  suc 
B  =  suc  B
)
108, 9eqtr4d 2498 . . . . 5  |-  ( Ord 
suc  B  ->  suc  U. suc  B  =  U. suc  suc 
B )
112, 3, 103syl 20 . . . 4  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. suc  B  =  U. suc  suc  B
)
12 unieq 4210 . . . . . 6  |-  ( A  =  suc  B  ->  U. A  =  U. suc  B )
13 suceq 4895 . . . . . 6  |-  ( U. A  =  U. suc  B  ->  suc  U. A  =  suc  U. suc  B
)
1412, 13syl 16 . . . . 5  |-  ( A  =  suc  B  ->  suc  U. A  =  suc  U.
suc  B )
15 suceq 4895 . . . . . 6  |-  ( A  =  suc  B  ->  suc  A  =  suc  suc  B )
1615unieqd 4212 . . . . 5  |-  ( A  =  suc  B  ->  U. suc  A  =  U. suc  suc  B )
1714, 16eqeq12d 2476 . . . 4  |-  ( A  =  suc  B  -> 
( suc  U. A  = 
U. suc  A  <->  suc  U. suc  B  =  U. suc  suc  B ) )
1811, 17syl5ibr 221 . . 3  |-  ( A  =  suc  B  -> 
( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  U. suc  A ) )
1918anabsi7 815 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  = 
U. suc  A )
20 eloni 4840 . . . 4  |-  ( A  e.  On  ->  Ord  A )
21 ordunisuc 6556 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
2220, 21syl 16 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
2322adantr 465 . 2  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  U. suc  A  =  A )
2419, 23eqtrd 2495 1  |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   U.cuni 4202   Ord word 4829   Oncon0 4830   suc csuc 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836
This theorem is referenced by:  rankxplim3  8203  rankxpsuc  8204
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